This story is built on the ridiculous humor of a farce. A farcical story is one in which the characters get involved in silly or unlikely situations. The hypochondriacal ramblings of the three friends are part of the farce. We will have to dig a little deeper to know whether George and William Samuel Harris are or are not truly hypochondriacs (I doubt it), but the exposition of the story gives us proof that J. is more the farcical social buffoon than the hypochondriac.
I sat for awhile, frozen with horror;.... I came to typhoid fever—read the symptoms—discovered that I had typhoid fever, must have had it for months without knowing it—wondered what else I had got; ... [I] began to get interested in my case, and determined to sift it to the bottom, and so started alphabetically .... Cholera I had, with severe complications; and diphtheria I seemed to have been born with. I plodded conscientiously through the twenty-six letters, and the only malady I could conclude I had not got was housemaid’s knee.
J. may well be convinced by a circular that his liver "was out of order," but he certainly cannot be convinced of his having all the diseases he reads about while at the British Museum, although it may well be true, as he says, that he at that time needed a treatment for hay fever: "I remember going to the British Museum one day to read up the treatment for some slight ailment of which I had a touch—hay fever, I fancy it was." Many people do suffer hay fever who are not at all hypochondriacal. Taking the last two diseases J. mentions, diphtheria and cholera, anyone with either of these (1) would be extremely contagious and (2) would be violently ill and (3) would often be rather dead, so they would surely know beyond anxious speculation (as would all and everyone around him) that they were ill.
In 1889, when this work was published, a child contracting diphtheria had only a 50 percent chance of recovering from it. So for J. to "have been born with" diphtheria is a pretty sure passport to a more ephemeral existence where disease isn't a worry or at best a passport to hospital quarantine until resolved, one way or the other. A diphtheria vaccine was introduced in the U.S. in 1920, which radically reduced the incidence and death rate, but in 1889, chances of contracting and surviving diphtheria were grim. The CDC says of adults and children that "up to half the people who got the disease died from it." Cholera is an even more contagious and deadly disease. The CDC says that "death can occur within hours" when left untreated.
That these are disease that are difficult to be hypochondriacal about (as they have obvious symptoms and are usually fatal) suggests that J. is constructed as a farcical narrator to humorously pursue the characterization of a pseudo-hypochondriac in this farcical story that comments on upper class English society. He is not really a hypochondriac.
Hypochondria is a troubling anxiety disorder that, like all anxiety disorders, is distressing and socially debilitating for the sufferer. A hypochondriac would not be able to wrap his worry and anxiety about disease in humor, farce and irreverence the way J. does. J. and his friends are more affected by ennui than by hypochondria. A ridiculous farcical character is meant to point the finger of sarcasm and satire, while using ironic language, at a social concept or social foible (i.e., a foolish idea or foolish habit). The social foible in Three Men is that of endlessly discussing medical conditions when boredom, or ennui, is the real problem. My opinion is that, no, J. is not really a hypochondriac. He is a narrator who is having a good laugh at himself while pointing the finger at the portion of society that shares his ennui and foibles.
George and Harris and Montmorency are not poetic ideals, but things of flesh and blood—especially George, who weighs about twelve stone. Other works may excel this in depth of thought and knowledge of human nature: other books may rival it in originality and size; but, for hopeless and incurable veracity, nothing yet discovered can surpass it. (Preface, London, August, 1889.)